Some localization theorems using a majorization technique

In this note we localize ordered real numbers through their upper and lower bounds solving a class of nonlinear optimization problems. To this aim, a majorization technique, which involves Schur-convex functions, has been applied and maximum and minimum elements of suitable sets are considered. The bounds we develop can be expressed in terms of the mean and higher centered moments of the number distribution. Meaningful results are obtained for real eigenvalues of a matrix of order n. Finally, numerical examples are provided, showing how former results in the literature can be sometimes improved through those methods